Reference request for a general-to-specific text(book) on abstract dynamical systems

Question

In all references on dynamical systems---encyclopedias, textbooks and articles---I have so far consulted, either

• there is from the beginning an emphasis on a certain class of dynamical system being studied (e.g. smooth dynamical systems or group actions), or
• the text is written very "high-level", without many concrete examples that illustrate the theory and without technically formulated theorems that carefully outline the various assumptions
• various viewpoints (dynamical vs. ergodic) and the definitions surrounding them aren't reconciled resp. contrasted (assuming they are introduced at all, such as a measure dynamical system)

Do you know of a text (ideally an advanced textbook) that does not suffer from these issues?

That would be a textbook that presents the theory as much as is possible in most general terms (meaning it develops the theory for dynamical systems $(G,X,\phi)$, where $G$ is a semigroup, $X$ a set and $\phi$ the evolution operator subject to the usual axioms) and then indicates exactly what further assumptions need to be made on $G$ and/or $X$ in order to obtain more specific theorems?

The first chapter of the first volume of Hasselblatt & Katok's "Handbook of Dynamical Systems" would come close, but unfortunately it is written (as is appropriate for an encyclopedic text) rather high-level with too few concrete examples to be helpful to me.
Something analogous to John Lee's trilogy of textbooks, "Introduction to Topological/Smooth/Riemannian Manifolds" would be what I'm looking for (ok, the last of the three book actually has a slightly different title, but let's not be pedantic :)), where one can first read up the theory for the general case (topological manifolds) and afterwards see in which direction the theory develops if one adds more structure (smoothnes, Riemannian metric).
Although the text I'm looking does not have to go that much into specialize topics, as Lee's books do, I would also be happy with a text (article, conference proceedings etc.) that stays with the foundational definitions and basic concepts and really clarifies the various relationships between them.

Answer 1

This is just a long comment. There are a couple of things I don't fully understand in the question, and perhaps that's the reason why it is a bit difficult to deal with (as the absence of answers testifies).

1. You write: "there is from the beginning an emphasis on a certain class of dynamical system being studied (e.g. smooth dynamical systems or group actions)". The problem for me here is the examples you give. Talking about group (and semigroup) actions is the most general/abstract point of view in dynamical systems. If even that is too particular for you, then it's difficult to do better.

2. Most of the literature on topological dynamics makes very simple assumptions once and for all: a (usually compact) metric space, a continuous functions, the action of $\mathbb{N}$ (or $\mathbb{Z}$). Any good book on the subject would be therefore a very well defined chapter within your desired general reference text. My two pennies: look at Topological and symbolic dynamics by Kurka.

What's wrong with composing yourself your "big text" gluing together a suitable number of books which are slightly more specific? Isn't it usual when dealing with such a gigantic subject? Would you search for an analogous book on, say, "geometry"?

3. Quoting again: "various viewpoints (dynamical vs. ergodic) and the definitions surrounding them aren't reconciled resp. contrasted". There are various excellent books in which, for instance, aspects concerning topological and measurable dynamics are carefully compared, but, again, they usually focus on one aspect (or a few aspects) rather than providing the most general theory possible. You can for instance look at this very nice book by Downarowicz, dealing with entropy (from both points of view). It may be useful in your perspective as the author tries in general to impose as little structure as possible on the phase space. For instance, the entropy structure is first presented considering the abstract theory of convergence of nets defined on abstract sets.